Expand knowledge base

1 files changed, 12 insertions, 15 deletions

diff --git a/source/know/concept/drude-model/index.md b/source/know/concept/drude-model/index.md
index 0026d90..8fcd7fb 100644
--- a/source/know/concept/drude-model/index.md
+++ b/source/know/concept/drude-model/index.md
@@ -11,7 +11,7 @@ layout: "concept"
 
 The **Drude model**, also known as
 the **Drude-Lorentz model** due to its analogy
-to the [Lorentz oscillator model](/know/concept/lorentz-oscillator-model/)
+to the [Lorentz oscillator model](/know/concept/lorentz-oscillator-model/),
 classically predicts the [dielectric function](/know/concept/dielectric-function/)
 and electric conductivity of a gas of free charges,
 as found in metals and doped semiconductors.
@@ -33,17 +33,16 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Where $$m$$ and $$q < 0$$ are the mass and charge of the electron.
-The first term is Newton's third law,
+The first term is Newton's second law,
 and the last term represents a damping force
 slowing down the electrons at rate $$\gamma$$.
 
 Inserting the ansatz $$\vb{x}(t) = \vb{x}_0 e^{- i \omega t}$$
-and isolating for the displacement $$\vb{x}$$, we find:
+and isolating for the amplitude $$\vb{x}_0$$, we find:
 
 $$\begin{aligned}
-    \vb{x}(t)
-    = \vb{x}_0 e^{- i \omega t}
-    = - \frac{q \vb{E}}{m (\omega^2 + i \gamma \omega)}
+    \vb{x}_0
+    = - \frac{q \vb{E}_0}{m (\omega^2 + i \gamma \omega)}
 \end{aligned}$$
 
 The polarization density $$\vb{P}(t)$$ is therefore as shown below.
@@ -76,8 +75,10 @@ leading to so-called **plasma oscillations** of the electron density
 (see also [Langmuir waves](/know/concept/langmuir-waves/)):
 
 $$\begin{aligned}
-    \varepsilon_r(\omega)
-    = 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega}
+    \boxed{
+        \varepsilon_r(\omega)
+        = 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega}
+    }
     \qquad\qquad
     \boxed{
         \omega_p
@@ -96,7 +97,7 @@ then we can identify three distinct scenarios for $$\varepsilon_r$$ here:
     allowing for self-sustained plasma oscillations.
 *   $$\omega > \omega_p$$, so $$\varepsilon_r > 0$$,
     so the index $$\sqrt{\varepsilon}$$ is real and asymptotically goes to $$1$$,
-    leading to high transparency and low reflectivity from air.
+    leading to high transparency and low reflectivity (coming from air).
 
 For most metals $$\omega_p$$ is ultraviolet,
 which explains why they typically appear shiny to us.
@@ -158,12 +159,8 @@ the dielectric function $$\varepsilon_r(\omega)$$ can be written as:
 
 $$\begin{aligned}
     \boxed{
-        \begin{aligned}
-            \varepsilon_r(\omega)
-            &= 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega}
-            \\
-            &= 1 + \frac{i \sigma(\omega)}{\varepsilon_0 \omega}
-        \end{aligned}
+        \varepsilon_r(\omega)
+        = 1 + \frac{i \sigma(\omega)}{\varepsilon_0 \omega}
     }
 \end{aligned}$$